Solve for $x$ : $ 6|x - 2| - 6 = -3|x - 2| + 5 $
Add $ {3|x - 2|} $ to both sides: $ \begin{eqnarray} 6|x - 2| - 6 &=& -3|x - 2| + 5 \\ \\ { + 3|x - 2|} && { + 3|x - 2|} \\ \\ 9|x - 2| - 6 &=& 5 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 9|x - 2| - 6 &=& 5 \\ \\ { + 6} &=& { + 6} \\ \\ 9|x - 2| &=& 11 \end{eqnarray} $ Divide both sides by ${9}$ $ \dfrac{9|x - 2|} {{9}} = \dfrac{11} {{9}} $ Simplify: $ |x - 2| = \dfrac{11}{9}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 2 = -\dfrac{11}{9} $ or $ x - 2 = \dfrac{11}{9} $ Solve for the solution where $x - 2$ is negative: $ x - 2 = -\dfrac{11}{9} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& -\dfrac{11}{9} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& -\dfrac{11}{9} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $9$ $ x = - \dfrac{11}{9} {+ \dfrac{18}{9}} $ $ x = \dfrac{7}{9} $ Then calculate the solution where $x - 2$ is positive: $ x - 2 = \dfrac{11}{9} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& \dfrac{11}{9} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& \dfrac{11}{9} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $9$ $ x = \dfrac{11}{9} {+ \dfrac{18}{9}} $ $ x = \dfrac{29}{9} $ Thus, the correct answer is $x = \dfrac{7}{9} $ or $x = \dfrac{29}{9} $.